%I #20 Jan 01 2024 08:00:20

%S 1,2,2,2,2,3,2,2,2,3,2,3,1,3,4,2,1,3,1,3,3,4,1,3,2,2,2,3,1,6,1,2,3,2,

%T 3,3,1,2,2,3,1,5,1,4,4,2,1,3,2,3,2,2,1,3,3,4,2,2,1,6,1,2,3,2,2,5,1,2,

%U 2,4,1,3,1,2,4,2,4,3,1,3,2,2,1,5,2,2,2,4,1,6

%N Number of distinct partition numbers dividing n.

%H Giovanni Resta, <a href="/A236103/b236103.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Amiram Eldar_, Jan 01 2024: (Start)

%F a(n) = Sum_{d|n} A167392(d).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A078506 = 2.510597... . (End)

%e For n = 20 the divisors of 20 are 1, 2, 4, 5, 10, 20 and three of them are also partition numbers: 1, 2, 5, so a(20) = 3.

%e For n = 42 the divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42 and five of them are also partition numbers: 1, 2, 3, 7, 42, so a(42) = 5.

%t p = {1}; Table[If[n >= Last@p, AppendTo[p, PartitionsP[1 + Length@p]]]; Length@Select[p, Mod[n, #] == 0 &], {n, 90}] (* _Giovanni Resta_, Jan 22 2014 *)

%Y Cf. A000041, A001221, A049575, A078506, A085543, A167392, A236102, A236105, A236107, A236108.

%K nonn

%O 1,2

%A _Omar E. Pol_, Jan 21 2014